Torsion functions on moduli spaces in view of the cluster algebra

We introduce non-acyclic $PGL_n(\mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Porti's $PGL_2(\mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(\mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichm\"{u}ler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.